Local Triangulation Research Articles

Correction of Chinese Dance Training Movements Based on Digital Feature Recognition Technology

In order to improve the effect of Chinese dance training, this paper combines digital feature recognition technology to correct and analyze Chinese dance training movements and constructs an intelligent auxiliary training system. In order to solve the travel time problem at the computationally complex interface and ensure its computational accuracy, a local adaptive triangulation technique is used in the fast-advance algorithm. Moreover, this paper designs the function of the system according to the user's needs, transforms the design concept into a figurative visual representation through the interactive prototype according to the function, carries out the visual design of the interface according to the interactive prototype, and uses the interactive technology to realize the development of the system. From the test analysis results, it can be seen that the Chinese dance training action correction system based on the digital feature recognition technology proposed in this paper has a good effect and can effectively promote the improvement of the Chinese dance training effect.

C-TOL: Convex triangulation for optimal node localization with weighted uncertainties

There is always a need to reduce localization error in any wireless sensor network (WSN), and our aim is to observe the impact of localization uncertainty on network awareness. When nodes are deployed in a 2D plane and their l2-norm ranged triangulations are found, usually the unweighted localization uncertainty values become absurdly large with large triangulation cases. Moreover, there is no regard for the disparity between the lengths of any two links on the localization uncertainty. The upper bound of uncertainty keeps on rising with formation of asymmetric node triangulations with longer internodal distances and sharper vertices. To address this gap, a convex combination weighted approach (C-TOL, standing for Convex-Triangulation for Optimal node Localization) for solving the localization uncertainty problem is described here. The advantage of the proposed method is shown with the help of rigorous mathematical analysis of weighted uncertainty behaviour. The relationship of sensor node symmetry with triangulation uncertainty is formulated algebraically by considering both symmetric as well as asymmetric triangulations. Cramer Rao bound is derived to justify estimation under triangulation uncertainty. This approach paves the way for the WSN to prioritize different kinds of triangulations. Numerical results reveal that the weighted method prefers triangulations with more symmetry; hence it consistently achieves significantly lower values of mean and standard deviations than the existing unweighted localization technique, especially for densely connected sensor networks. Moreover, the proposed method shows robust localization performance for sparsely deployed networks as well, when compared to recent methods in literature.

Positive geometry, local triangulations, and the dual of the Amplituhedron

We initiate the systematic study of local positive spaces which arise in the context of the Amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory. We show that all local positive spaces relevant for one-loop MHV amplitudes are characterized by certain sign-flip conditions and are associated with surprisingly simple logarithmic forms. In the maximal sign-flip case they are finite one-loop octagons. Particular combinations of sign-flip spaces can be glued into new local positive geometries. These correspond to local pentagon integrands that appear in the local expansion of the MHV one-loop amplitude. We show that, geometrically, these pentagons do not triangulate the original Amplituhedron space but rather its twin “Amplituhedron-Prime”. This new geometry has the same boundary structure as the Amplituhedron (and therefore the same logarithmic form) but differs in the bulk as a geometric space. On certain two-dimensional boundaries, where the Amplituhedron geometry reduces to a polygon, we check that both spaces map to the same dual polygon. Interestingly, we find that the pentagons internally triangulate that dual space. This gives a direct evidence that the chiral pentagons are natural building blocks for a yet-to-be discovered dual Amplituhedron.

Hierarchical Modeling of Street Trees Using Mobile Laser Scanning

This paper proposes a novel method to reconstruct hierarchical 3D tree models from Mobile Laser Scanning (MLS) point clouds. Starting with a neighborhood graph from the tree point clouds, the method treats the root point of the tree as a source point and determines an initial tree skeleton by using the Dijkstra algorithm. The initial skeleton lines are then optimized by adjusting line connectivity and branch nodes based on morphological characteristics of the tree. Finally, combined with the tree point clouds, the radius of each branch skeleton node is estimated and flat cones are used to simulate tree branches. A local triangulation method is used to connect the gaps between two joint flat cones. Demonstrated by street trees of different sizes and point densities, the proposed method can extract street tree skeletons effectively, generate tree models with higher fidelity, and reconstruct trees with different details according to the skeleton level. It is found out the tree modeling error is related to the average point spacing, with a maximum error at the coarsest level 6 being about 0.61 times the average point spacing. The main source of the modeling error is the self-occlusion of trees branches. Such findings are both theoretically and practically useful for generating high-precision tree models from point clouds. The developed method can be an alternative to the current ones that struggle to balance modeling efficiency and modeling accuracy.

The Internet-of-Battlefield-Things (IoBT)-Based Enemy Localization Using Soldiers Location and Gunshot Direction

The real-time information of enemy locations is capable to transform the outcome of combat operations. Such information gathered using connected soldiers on the Internet of Battlefield Things (IoBT) is highly beneficial to create situational awareness (SA) and to plan an effective war strategy. This article presents the novel enemy localization method that uses the soldier's own locations and their gunshot direction. The hardware prototype has been developed that uses a triangulation for an enemy localization in two soldiers and a single enemy scenario. 4.24±1.77 m of average localization error and ±4° of gunshot direction error has been observed during this prototype testing. This basic model is further extended using three-stage software simulation for multiple soldiers and multiple enemy scenarios with the necessary assumptions. The effective algorithm has been proposed, which differentiates between the ghost and true predictions by analyzing the groups of subsequent shooting intents (i.e., frames). Four different complex scenarios are tested in the first stage of the simulation, around three to six frames are required for the accurate enemy localization in the relatively simple cases, and nine frames are required for the complex cases. The random error within ±4° in gunshot direction is included in the second stage of the simulation which required almost double the number of frames for similar four cases. As the number of frames increases, the accuracy of the proposed algorithm improves and better ghost point elimination is observed. In the third stage, two conventional clustering algorithms are implemented to validate the presented work. The comparative analysis shows that the proposed algorithm is faster, computationally simple, consistent, and reliable compared with others. Detailed analysis of hardware and software results for various scenarios has been discussed in this article.

Unveiling the Hidden Rules of Spherical Viruses Using Point Arrays.

Since its introduction, the Triangulation number has been the most successful and ubiquitous scheme for classifying spherical viruses. However, despite its many successes, it fails to describe the relative angular orientations of proteins, as well as their radial mass distribution within the capsid. It also fails to provide any critical insight into sites of stability, modifications or possible mutations. We show how classifying spherical viruses using icosahedral point arrays, introduced by Keef and Twarock, unveils new geometric rules and constraints for understanding virus stability and key locations for exterior and interior modifications. We present a modified fitness measure which classifies viruses in an unambiguous and rigorous manner, irrespective of local surface chemistry, steric hinderance, solvent accessibility or Triangulation number. We then use these point arrays to explain the immutable surface loops of bacteriophage MS2, the relative reactivity of surface lysine residues in CPMV and the non-quasi-equivalent flexibility of the HBV dimers. We then explain how point arrays can be used as a predictive tool for site-directed modifications of capsids. This success builds on our previous work showing that viruses place their protruding features along the great circles of the asymmetric unit, demonstrating that viruses indeed adhere to these geometric constraints.

Conforming window functions for meshfree methods

Window functions provide a base for the construction of approximation functions in many meshfree methods. They control the smoothness and extent of the approximation functions and are commonly defined using Euclidean distances which helps eliminate the need for a meshed discretization, simplifying model development for some classes of problems. However, for problems with complicated geometries such as nonconvex or multi-body domains, poor solution accuracy and convergence can occur unless the extents of the window functions, and thus approximation functions, are carefully controlled, often a time consuming or intractable task. In this paper, we present a method to provide more control in window function design, allowing efficient and systematic handling of complex geometries. “Conforming” window functions are constructed using Bernstein–Bézier splines defined on local triangulations with constraints imposed to control smoothness. Graph distances are used in conjunction with Euclidean metrics to provide adequate information for shaping the window functions. The conforming window functions are demonstrated using the Reproducing Kernel Particle Method showing improved accuracy and convergence rates for problems with challenging geometries. Conforming window functions are also demonstrated as a means to simplify the imposition of essential boundary conditions.

A lifted local Galerkin method for solving the reaction–diffusion equations on implicit surfaces

The reaction–diffusion equations models on surfaces have received growing interest and have been applied to simulate the physical processes and chemical reactions on ultra-thin materials, solid surfaces and biological films. In this article, we present a lifted local Galerkin method for solving the reaction–diffusion equations on implicit surfaces. The basic idea of proposed method is to discretize the equations on tangent plane of each surface node. Based on this idea, for each surface node, we first construct the local triangulation mesh via using this node and lifted points of its neighborhood on the tangent plane for this node. After setting a constant extension of surface data between each neighborhood and lifted point, we construct the discrete scheme by a local Galerkin method. Finally, the convergence tests and some applications in engineering science of the proposed method are shown to verify the feasibility and correction.

Joint tomographic inversion of first-arrival and reflection traveltimes for recovering 2-D seismic velocity structure with an irregular free surface

Irregular surface flattening, which is based on a boundary conforming grid and the transformation between curvilinear and Cartesian coordinate systems, is a mathematical method that can elegantly handle irregular surfaces, but has been limited to obtaining first arrivals only. By combining a multistage scheme with the fast-sweeping method (FSM, the method to obtain first-arrival traveltime in curvilinear coordinates), the reflected waves from a crustal interface can be traced in a topographic model, in which the reflected wave-front is obtained by reinitializing traveltimes in the interface for upwind branches. A local triangulation is applied to make a connection between velocity and interface nodes. Then a joint inversion of first-arrival and reflection traveltimes for imaging seismic velocity structures in complex terrains is presented. Numerical examples all perform well with different seismic velocity models. The increasing topographic complexity and even use of a high curvature reflector in these models demonstrate the reliability, accuracy and robustness of the new working scheme; checkerboard testing illustrates the method’s high resolution. Noise tolerance testing indicates the method’s ability to yield practical traveltime tomography. Further development of the multistage scheme will allow other later arrivals to be traced and used in the traveltime inversion.

FE-SViT

As a promising bio-cryptographic technique, the fuzzy extractor seamlessly binds biometrics and cryptography for template protection and key generation. However, most existing methods hardly solve the following issues simultaneously: (1) Fingerprint registration, (2) Verification accuracy, (3) Security strength, and (4) Computational efficiency. In this article, we introduce a bio-crypto-oriented fingerprint verification scheme - Selective Vertex-indexed Triangulation (SViT) which maps minutia global topology to local triangulation with minimum information loss. Then, a SViT-based fuzzy extractor framework (FE-SViT) is proposed and high verification accuracy is achieved. The FE-SViT is highly parallelizable and efficient which makes it suitable for embedded devices.

Utilizing the Virtual Triangulation for Wireless Indoor Localization of Mobile Devices with Channel State Information

To locate an RF device or to enable target oriented communication services in wireless indoor environment, RF properties as RSSI and CSI have been widely adopted in the research and commercial societies. However, the complex indoor scenario makes the task to locate an RF device with fine-grained accuracy more difficult. This paper deals with a virtual triangulation utilization to estimate the length of Line of Sight (LoS) signal out of non LoS on receiving multipath signals from any dimension. Thus, it forms a virtual right angled triangle of the received signals. From the angles ratio, the propose method estimates the side corresponding to the reference LoS length. Eventually, it estimates the distance error between the measured reference LoS of the virtual triangle and the approximated LoS from the CSI. The error is utilized to provide a basement for more precise localization of an RF device.

A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations

We present a high-order accurate space–time discontinuous Galerkin method for solving two-dimensional compressible flow problems on fully unstructured space–time meshes. The discretization is based on a nodal formulation, with appropriate numerical fluxes for the first and the second-order terms, respectively. The scheme is implicit, and we solve the resulting non-linear systems using a parallel Newton–Krylov solver. The meshes are produced by a mesh moving technique with element connectivity updates, and the corresponding space–time elements are produced directly based on these local operations. To obtain globally conforming tetrahedral meshes, we first derive the required conditions on a prism boundary mesh to allow for a valid local triangulation. Next, we present an efficient algorithm for finding a global mesh that satisfies these conditions. We also show how to add and remove mesh nodes, again using local constructs for the space–time mesh. Our method is demonstrated on a number of test problems, showing the high-order accuracy for model problems, and the ability to solve flow problems on domains with complex large deformations.

TH‐E‐BRE‐01: A 3D Solver of Linear Boltzmann Transport Equation Based On a New Angular Discretization Method with Positivity for Photon Dose Calculation Benchmarked with Geant4

Purpose:The Linear Boltzmann Transport Equation (LBTE) solved through statistical Monte Carlo (MC) method provides the accurate dose calculation in radiotherapy. This work is to investigate the alternative way for accurately solving LBTE using deterministic numerical method due to its possible advantage in computational speed from MC.Methods:Instead of using traditional spherical harmonics to approximate angular scattering kernel, our deterministic numerical method directly computes angular scattering weights, based on a new angular discretization method that utilizes linear finite element method on the local triangulation of unit angular sphere. As a Result, our angular discretization method has the unique advantage in positivity, i.e., to maintain all scattering weights nonnegative all the time, which is physically correct. Moreover, our method is local in angular space, and therefore handles the anisotropic scattering well, such as the forward‐peaking scattering. To be compatible with image‐guided radiotherapy, the spatial variables are discretized on the structured grid with the standard diamond scheme. After discretization, the improved sourceiteration method is utilized for solving the linear system without saving the linear system to memory. The accuracy of our 3D solver is validated using analytic solutions and benchmarked with Geant4, a popular MC solver.Results:The differences between Geant4 solutions and our solutions were less than 1.5% for various testing cases that mimic the practical cases. More details are available in the supporting document.Conclusion:We have developed a 3D LBTE solver based on a new angular discretization method that guarantees the positivity of scattering weights for physical correctness, and it has been benchmarked with Geant4 for photon dose calculation.

A local mesh method for solving PDEs on point clouds

In this work, we introduce a numerical method to approximate differential operators and integrals on point clouds sampled from a two dimensional manifold embedded in $\mathbb{R}^n$. Global mesh structure is usually hard to construct in this case. While our method only relies on the local mesh structure at each data point, which is constructed through local triangulation in the tangent space obtained by local principal component analysis (PCA). Once the local mesh is available, we propose numerical schemes to approximate differential operators and define mass matrix and stiffness matrix on point clouds, which are utilized to solve partial differential equations (PDEs) and variational problems on point clouds. As numerical examples, we use the proposed local mesh method and variational formulation to solve the Laplace-Beltrami eigenproblem and solve the Eikonal equation for computing distance map and tracing geodesics on point clouds.

INTEGRATION OF HETEROGENOUS DIGITAL SURFACE MODELS

Abstract. The application of extended digital surface models often reveals, that despite an acceptable global accuracy for a given dataset, the local accuracy of the model can vary in a wide range. For high resolution applications which cover the spatial extent of a whole country, this can be a major drawback. Within the Swiss National Forest Inventory (NFI), two digital surface models are available, one derived from LiDAR point data and the other from aerial images. Automatic photogrammetric image matching with ADS80 aerial infrared images with 25cm and 50cm resolution is used to generate a surface model (ADS-DSM) with 1m resolution covering whole switzerland (approx. 41000 km2). The spatially corresponding LiDAR dataset has a global point density of 0.5 points per m2 and is mainly used in applications as interpolated grid with 2m resolution (LiDAR-DSM). Although both surface models seem to offer a comparable accuracy from a global view, local analysis shows significant differences. Both datasets have been acquired over several years. Concerning LiDAR-DSM, different flight patterns and inconsistent quality control result in a significantly varying point density. The image acquisition of the ADS-DSM is also stretched over several years and the model generation is hampered by clouds, varying illumination and shadow effects. Nevertheless many classification and feature extraction applications requiring high resolution data depend on the local accuracy of the used surface model, therefore precise knowledge of the local data quality is essential. The commercial photogrammetric software NGATE (part of SOCET SET) generates the image based surface model (ADS-DSM) and delivers also a map with figures of merit (FOM) of the matching process for each calculated height pixel. The FOM-map contains matching codes like high slope, excessive shift or low correlation. For the generation of the LiDAR-DSM only first- and last-pulse data was available. Therefore only the point distribution can be used to derive a local accuracy measure. For the calculation of a robust point distribution measure, a constrained triangulation of local points (within an area of 100m2) has been implemented using the Open Source project CGAL. The area of each triangle is a measure for the spatial distribution of raw points in this local area. Combining the FOM-map with the local evaluation of LiDAR points allows an appropriate local accuracy evaluation of both surface models. The currently implemented strategy ("partial replacement") uses the hypothesis, that the ADS-DSM is superior due to its better global accuracy of 1m. If the local analysis of the FOM-map within the 100m2 area shows significant matching errors, the corresponding area of the triangulated LiDAR points is analyzed. If the point density and distribution is sufficient, the LiDAR-DSM will be used in favor of the ADS-DSM at this location. If the local triangulation reflects low point density or the variance of triangle areas exceeds a threshold, the investigated location will be marked as NODATA area. In a future implementation ("anisotropic fusion") an anisotropic inverse distance weighting (IDW) will be used, which merges both surface models in the point data space by using FOM-map and local triangulation to derive a quality weight for each of the interpolation points. The "partial replacement" implementation and the "fusion" prototype for the anisotropic IDW make use of the Open Source projects CGAL (Computational Geometry Algorithms Library), GDAL (Geospatial Data Abstraction Library) and OpenCV (Open Source Computer Vision).

Treatment of Checkerboard Pressure in the Collocated Unstructured Finite-Volume Scheme

In this article, a new concept, “local triangle,” is described. A smoother pressure field is then obtained through the utilization of local triangles in a multiblock local triangulation method. The cell face velocity is corrected with a pressure-correction term which is usually underrelaxed. It is demonstrated that this relaxation factor has to be one, and it cannot be the general relaxation factor. It is indicated that the checkerboard oscillations are more significant at higher cell Reynolds numbers and, in turn, in this article, numerical simulations are mainly performed in turbulent flow.

Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.

Dynamic shape transformations of fluid vesicles

We incorporate a volume-control algorithm into a recently developed one-particle-thick mesoscopic fluid membrane model to study vesicle shape transformation under osmotic conditions. Each coarse-grained particle in the model represents a cluster of lipid molecules and the inter-particle interaction potential effectively captures the dual character of fluid membranes as elastic shells with out-of-plane bending rigidity and 2D viscous fluids with in-plane viscosity. The osmotic pressure across the membrane is accounted for by an external potential, where the instantaneous volume of the vesicles is calculated via a local triangulation algorithm. Through coarse-grained molecular dynamics simulations, we mapped out a phase diagram of the equilibrium vesicle shapes in the space of spontaneous curvature and reduced vesicle volume. The produced equilibrium vesicle shapes agree strikingly well with previous experimental data. We further found that the vesicle shape transformation pathways depend on the volume change rate of the vesicle, which manifests the role of dynamic relaxation of internal stresses in vesicle shape transformations. Besides providing an efficient numerical tool for the study of membrane deformations, our simulations shed light on eliciting desired cellular functions via experimental control of membrane configurations.

Consistent triangulation for mobile robot localization using discontinuous angular measurements

Localization is a fundamental operation for the navigation of mobile robots. The standard localization algorithms fuse external measurements of the environment with the odometric evolution of the robot pose to obtain its optimal estimation. In this work, we present a different approach to determine the pose using angular measurements discontinuously obtained in time. The presented method is based on an Extended Kalman Filter (EKF) with a state-vector composed of the external angular measurements. This algorithm keeps track of the angles between actual measurements from robot odometric information. This continuous angular estimation allows the consistent use of the triangulation methods to determine the robot pose at any time during its motion. The article reports experimental results that show the localization accuracy obtained by means of the presented approach. These results are compared to the ones obtained applying the EKF algorithm with the standard pose state-vector. For the experiments, an omnidirectional robotic platform with omnidirectional wheels is used.

Description of a method for localizing swarming mosquitoes and other insects in 3D space with visualizations

AbstractMale mosquitoes swarm to find mates, but the characteristics of these swarms have not yet been quantitatively analyzed in the field. This paper describes a simple method used recently to localize individual Anopheles gambiae in a swarm via stereoscopic image analysis of video footage swarms recorded in Doneguebougou, Mali. The methods described here represent a streamlining of simple triangulation for insect localization in the hope that other researchers will be able to apply the method for studying fundamental questions about flying insects or other taxa.